A review of finite size effects in quasi-zero dimensional superconductors

Quantum confinement and surface effects (SEs) dramatically modify most solid state phenomena as one approaches the nanometer scale, and superconductivity is no exception. Though we may expect significant modifications from bulk superconducting properties when the system dimensions become smaller than the characteristic length scales for bulk superconductors-such as the coherence length or the penetration depth-it is now established that there is a third length scale which ultimately determines the critical size at which Cooper pairing is destroyed. In quasi-zero-dimensional (0D) superconductors (e.g. nanocrystalline materials, isolated or embedded nanoparticles), one may define a critical particle diameter below which the mean energy level spacing arising from quantum confinement becomes equal to the bulk superconducting energy gap. The so-called Anderson criterion provides a remarkably accurate estimate of the limiting size for the destabilization of superconductivity in nanosystems. This review of size effects in quasi-0D superconductors is organized as follows. A general summary of size effects in nanostructured superconductors (section 1) is followed by a brief overview of their synthesis (section 2) and characterization using a variety of techniques (section 3). Section 4 reviews the size-evolution of important superconducting parameters-the transition temperature, critical fields and critical current-as the Anderson limit is approached from above. We then discuss the effect of thermodynamic fluctuations (section 5), which become significant in confined systems. Improvements in fabrication methods and the increasing feasibility of addressing individual nanoparticles using scanning probe techniques have lately opened up new directions in the study of nanoscale superconductivity. Section 6 reviews both experimental and theoretical aspects of the recently discovered phenomena of 'parity effect' and 'shell effect' that lead to a strong, non-monotonic size dependence of the superconducting energy gap and associated properties. Finally, we discuss in section 7 the properties of ordered heterostructures (bilayers and multilayers of alternating superconducting and normal phases) and disordered heterostructures (nanocomposites consisting of superconducting and normal phases), which are primarily controlled by the proximity effect.